| Title:
|
On continuous self-maps and homeomorphisms of the Golomb space (English) |
| Author:
|
Banakh, Taras |
| Author:
|
Mioduszewski, Jerzy |
| Author:
|
Turek, Sławomir |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
59 |
| Issue:
|
4 |
| Year:
|
2018 |
| Pages:
|
423-442 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The Golomb space ${\mathbb N}_\tau$ is the set ${\mathbb N}$ of positive integers endowed with the topology $\tau$ generated by the base consisting of arithmetic progressions $\{a+ bn: n\ge 0\}$ with coprime $a,b$. We prove that the Golomb space ${\mathbb N}_\tau$ has continuum many continuous self-maps, contains a countable disjoint family of infinite closed connected subsets, the set $\Pi$ of prime numbers is a dense metrizable subspace of ${\mathbb N}_\tau$, and each homeomorphism $h$ of ${\mathbb N}_\tau$ has the following properties: $h(1)=1$, $h(\Pi)=\Pi$, $\Pi_{h(x)}=h(\Pi_x)$, and $h(x^{{\mathbb N}})=h(x)^{\,\mathbb N}$ for all $x\in{\mathbb N}$. Here $x^{\mathbb N}:=\{x^n\colon n\in{\mathbb N}\}$ and $\Pi_x$ denotes the set of prime divisors of $x$. (English) |
| Keyword:
|
Golomb space |
| Keyword:
|
arithmetic progression |
| Keyword:
|
superconnected space |
| Keyword:
|
homeomorphism |
| MSC:
|
11A41 |
| MSC:
|
54D05 |
| idZBL:
|
Zbl 06997360 |
| idMR:
|
MR3914710 |
| DOI:
|
10.14712/1213-7243.2015.269 |
| . |
| Date available:
|
2018-12-28T15:06:27Z |
| Last updated:
|
2021-01-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147548 |
| . |
| Reference:
|
[1] Apostol T. M.: Introduction to Analytic Number Theory.Undergraduate Texts in Mathematics, Springer, New York, 1976. MR 0434929 |
| Reference:
|
[2] Artin E., Tate J.: Class Field Theory.Advanced Book Classics, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, 1990. MR 1043169 |
| Reference:
|
[3] Banakh T.: Is the Golomb countable connected space topologically rigid?.available at https://mathoverflow.net/questions/285557. |
| Reference:
|
[4] Banakh T.: A simultaneous generalization of the Grunwald-Wang and Dirichlet theorems on primes.available at https://mathoverflow.net/questions/310130. |
| Reference:
|
[5] Banakh T.: Is the identity function a unique multiplicative homeomorphism of $ \mathbb N$?.available at https://mathoverflow.net/questions/310163. |
| Reference:
|
[6] Brown M.: A countable connected Hausdorff space.Bull. Amer. Math. Soc. 59 (1953), Abstract 423, 367. |
| Reference:
|
[7] Clark P. L., Lebowitz-Lockard N., Pollack P.: A note on Golomb topologies.Quaest. Math. published online, available at https://doi.org/10.2989/16073606.2018.1438533. 10.2989/16073606.2018.1438533 |
| Reference:
|
[8] Dirichlet P. G. L.: Lectures on Number Theory.History of Mathematics, 16, American Mathematical Society, Providence, London, 1999. MR 1710911, 10.1090/hmath/016 |
| Reference:
|
[9] Engelking R.: General Topology.Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin, 1989. Zbl 0684.54001, MR 1039321 |
| Reference:
|
[10] Engelking R.: Theory of Dimensions Finite and Infinite.Sigma Series in Pure Mathematics, 10, Heldermann Verlag, Lemgo, 1995. Zbl 0872.54002, MR 1363947 |
| Reference:
|
[11] Furstenberg H.: On the infinitude of primes.Amer. Math. Monthly 62 (1955), 353. MR 0068566, 10.2307/2307043 |
| Reference:
|
[12] Gauss C. F.: Disquisitiones Arithmeticae.Springer, New York, 1986. Zbl 1167.11001 |
| Reference:
|
[13] Golomb S. W.: A connected topology for the integers.Amer. Math. Monthly 66 (1959), 663–665. MR 0107622, 10.1080/00029890.1959.11989385 |
| Reference:
|
[14] Golomb S. W.: Arithmetica topologica.General Topology and Its Relations to Modern Analysis and Algebra, Proc. Symp., Prague, 1961, Academic Press, New York; Publ. House Czech. Acad. Sci., Praha (1962), 179–186. MR 0154249 |
| Reference:
|
[15] Jones G. A., Jones J. M.: Elementary Number Theory.Springer Undergraduate Mathematics Series, Springer, London, 1998. MR 1610533 |
| Reference:
|
[16] Knaster B., Kuratowski K.: Sur les ensembles connexes.Fund. Math. 2 (1921), no. 1, 206–256 (French). 10.4064/fm-2-1-206-255 |
| Reference:
|
[17] Knopfmacher J., Porubský Š.: Topologies related to arithmetical properties of integral domains.Exposition Math. 15 (1997), no. 2, 131–148. MR 1458761 |
| Reference:
|
[18] Steen L. A., Seebach J. A., Jr.: Counterexamples in Topology.Dover Publications, Mineola, 1995. Zbl 0386.54001, MR 1382863 |
| Reference:
|
[19] Stevenhagen P., Lenstra H. W., Jr.: Chebotarëv and his density theorem.Math. Intelligencer 18 (1996), no. 2, 26–37. MR 1395088, 10.1007/BF03027290 |
| Reference:
|
[20] Sury B.: Frobenius and his density theorem for primes.Resonance 8 (2003), no. 12, 33–41. 10.1007/BF02839049 |
| Reference:
|
[21] Szczuka P.: The connectedness of arithmetic progressions in Furstenberg's, Golomb's, and Kirch's topologies.Demonstratio Math. 43 (2010), no. 4, 899–909. MR 2761648 |
| Reference:
|
[22] Szczuka P.: The Darboux property for polynomials in Golomb's and Kirch's topologies.Demonstratio Math. 46 (2013), no. 2, 429–435. MR 3098036 |
| Reference:
|
[23] Wang S.: A counter-example to Grunwald's theorem.Ann. of Math. (2) 49 (1948), 1008–1009. MR 0026992, 10.2307/1969410 |
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